65,473 research outputs found
On the Convex Feasibility Problem
The convergence of the projection algorithm for solving the convex
feasibility problem for a family of closed convex sets, is in connection with
the regularity properties of the family. In the paper [18] are pointed out four
cases of such a family depending of the two characteristics: the emptiness and
boudedness of the intersection of the family. The case four (the interior of
the intersection is empty and the intersection itself is bounded) is unsolved.
In this paper we give a (partial) answer for the case four: in the case of two
closed convex sets in R3 the regularity property holds.Comment: 14 pages, exposed on 5th International Conference "Actualities and
Perspectives on Hardware and Software" - APHS2009, Timisoara, Romani
Zero-Convex Functions, Perturbation Resilience, and Subgradient Projections for Feasibility-Seeking Methods
The convex feasibility problem (CFP) is at the core of the modeling of many
problems in various areas of science. Subgradient projection methods are
important tools for solving the CFP because they enable the use of subgradient
calculations instead of orthogonal projections onto the individual sets of the
problem. Working in a real Hilbert space, we show that the sequential
subgradient projection method is perturbation resilient. By this we mean that
under appropriate conditions the sequence generated by the method converges
weakly, and sometimes also strongly, to a point in the intersection of the
given subsets of the feasibility problem, despite certain perturbations which
are allowed in each iterative step. Unlike previous works on solving the convex
feasibility problem, the involved functions, which induce the feasibility
problem's subsets, need not be convex. Instead, we allow them to belong to a
wider and richer class of functions satisfying a weaker condition, that we call
"zero-convexity". This class, which is introduced and discussed here, holds a
promise to solve optimization problems in various areas, especially in
non-smooth and non-convex optimization. The relevance of this study to
approximate minimization and to the recent superiorization methodology for
constrained optimization is explained.Comment: Mathematical Programming Series A, accepted for publicatio
Alternating Projections and Douglas-Rachford for Sparse Affine Feasibility
The problem of finding a vector with the fewest nonzero elements that
satisfies an underdetermined system of linear equations is an NP-complete
problem that is typically solved numerically via convex heuristics or
nicely-behaved nonconvex relaxations. In this work we consider elementary
methods based on projections for solving a sparse feasibility problem without
employing convex heuristics. In a recent paper Bauschke, Luke, Phan and Wang
(2014) showed that, locally, the fundamental method of alternating projections
must converge linearly to a solution to the sparse feasibility problem with an
affine constraint. In this paper we apply different analytical tools that allow
us to show global linear convergence of alternating projections under familiar
constraint qualifications. These analytical tools can also be applied to other
algorithms. This is demonstrated with the prominent Douglas-Rachford algorithm
where we establish local linear convergence of this method applied to the
sparse affine feasibility problem.Comment: 29 pages, 2 figures, 37 references. Much expanded version from last
submission. Title changed to reflect new development
Application of projection algorithms to differential equations: boundary value problems
The Douglas-Rachford method has been employed successfully to solve many
kinds of non-convex feasibility problems. In particular, recent research has
shown surprising stability for the method when it is applied to finding the
intersections of hypersurfaces. Motivated by these discoveries, we reformulate
a second order boundary valued problem (BVP) as a feasibility problem where the
sets are hypersurfaces. We show that such a problem may always be reformulated
as a feasibility problem on no more than three sets and is well-suited to
parallelization. We explore the stability of the method by applying it to
several examples of BVPs, including cases where the traditional Newton's method
fails
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